Re: Small Markov models problem


David Jones wrote:
I think that most modellers would want to work with the conditional
probabilities and to use these to judge whether a model showed more or
less dependence than another. However, working with conditional
probabilities leaves the extra step of having to fix-up the marginal
distibutions to remain the same. For your model (treating the 0.3, 0.7
weights a derived from a parameter), can you derive the conditional
probabilities and show that that the parameter has the effect of
indicating more or less dependence in terms of the conditional
probabilities?

Thanks. If I think of it that way, I think I just made a silly mistake.
I think that I converted the representation of the Markov model to the
non-standard form for no reason. Maybe.

What I have are some Markov models,

p1( a | b )
p1( b | b )
p1( a | a )
p1( b | a )

p2( a | b )
p2( b | b )
p2( a | a )
p2( b | a )

and the long term state probabilties which are the same in both models:

p(a), p(b)

There are underlying variables used in calculating these models. One of
these is the relationship between two variables. In p1() these
variables were fully independent. In p2() these variables were
equivalent. I previously converted the network into a set of
probabilities of conjuncts as that allowed me to calculate simple
weighed sums between the probabilities to create Markov models
representing intermediate states between the two extremes, with the
fortuitous consequence that for all new such models, the long term
state probabilities p(a) and p(b) remained the same, as is necessary.

So, thinking about a weight w such that 0 <= w <= 1.0, I was
calculating:

p3( a & b ) = w * p1( a & b ) + (1.0-w) * p2( a & b ) [1]

Finally getting past the recap, I could simply write:

p3( a | b ) = w * p1( a | b ) + (1.0-w) * p2( a | b ) [2](*)

Expanding slightly:

p3( a | b ) = w * ( p1( a & b ) / p( a ) ) + (1.0-w) * ( p2( a & b ) /
p( a ) )

removing the common factor:

p3( a | b ) = ( w * p1( a & b ) + (1.0-w) * p2( a & b ) ) / p( a )

which by the previous definition of p3( a & b ) [1]

p3( a | b ) = p3( a & b ) / p( a )

So in other words, the representation of the Markov model as
probabilities of conjunctions is completely unnecessary, as for any
weight w, creating the interpolated Markov model p3 by using [2]:

p3( a | b ) = w * p1( a | b ) + (1.0-w) * p2( a | b ) [3]
[etc]

is exactly equivalent to what I did previously, and will create exactly
the same Markov model. Therefore, I was wrong in assuming that this
would not preserve the state probabilities p( a ), p( b ).

What I have above is not my problem, because my real networks have four
nodes and are calculated from assumptions about two Markov models
producing independent sequences. But I'm reasonably confident I can
work through the whole thing backwards from the analogue to [3] to show
that the intermediate models imply intermediate degrees of dependence
between the variables I'm interested in while keeping the values of the
other independent parameters constant.

(*) hope I'm not getting to pretentious here by numbering the
equations.

Cheers,

Ross-c

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